Induced voltages and Inductance Faraday s Law concept #1, 4, 5, 8, 13 Problem # 1, 3, 4, 5, 6, 9, 10, 13, 15, 24, 23, 25, 31, 32a, 34, 37, 41, 43, 51, 61
Last chapter we saw that a current produces a magnetic field. In 1831 experiments by Michael Faraday and Joseph Henry showed that a changing magnetic field could induce a current in a circuit. Faraday s setup. Switch Ammeter + _ Battery 0.0 ma
The coil with the switch is connected to a battery. (Primary coil) When current goes through a coil, it produces a magnetic field. The coils are wrapped around an iron ring to intensify the magnetic field. The secondary coil is hooked up to an ammeter. This coil is not hooked up to a battery.
When the switch in the primary coil is closed, the ammeter reads a current in the secondary coil for a short moment, then returns to zero. When the switch is opened, the ammeter momentarily measures a current in the opposite direction before returning to zero. When there is a steady current in the primary coil, there is no current read by the ammeter.
Conclusion: An electric current can be produced from a changing magnetic field. The current produced in the secondary coil occurs only for the instant the magnetic field through the secondary coil is changing. The secondary circuit behaves as though a source of emf (a battery) was connected to it for a short time. An induced emf is produced in the secondary circuit by the changing magnetic field.
emf electric motive force, not really a force. This is a source of electrical work/energy per unit charge. work/charge = volt Devices that increase the potential energy of circulating charges (batteries, generators) are sources of emf. Think of emf as a voltage increase.
While the magnetic field inside the secondary coil was changing, the secondary coil acted as is it was connected to a battery. The changing magnetic field induced an electric field in the secondary wire that caused the current to flow.
Changing magnetic fields induce electric fields. A constant magnetic field can also induce an electric field. Example of this is an electric generator. If the magnetic field is constant then what is changing? The property that creates an electric field is the changing of the magnetic flux.
Magnetic flux Magnetic flux is defined in the same way electric flux was defined earlier. Magnetic flux through a loop, is proportional to the strength of the B-field passing through the plane of the loop and the area of the loop. B = B A cos B cos is the component of the B-field that is perpendicular to the loop. B side view of loop
Magnetic flux The value of the magnetic flux is proportional to the number of B-field lines passing through the loop. B = B A cos is maximized when = 0. This is when the B-field is perpendicular to the loop. see fig. 20.3 for maximizing/minimizing the flux work example 20.1
Faraday s law of Induction Consider a wire loop connected to an ammeter. Moving a magnet towards the loop will induce a current in one direction. When the magnet is stationary, there is no induced current. Moving the magnet away from the loop induces a current in the opposite direction. This is similar to the Faraday experiment shown earlier.
Faraday s Law of induction An emf is induced in a circuit when the through the circuit changes with time. B The instantaneous emf induced in a circuit equals the negative rate of change of B with respect to time. This is Faraday s Law of magnetic induction. N t B
Lenz s Law: The induced current travels in the direction that creates a magnetic field with flux opposing the change in the original flux through the circuit. If the flux is increasing in one direction, the induced current will be in the direction so that its own magnetic flux will be in the direction opposite of the original flux. Nature wants to keep the flux constant.
The induced magnetic flux does not have to be in the opposite direction of the original flux. Fig. 20.5. The original flux is upwards. As the B-field is reduced in strength, the flux is reduced. Lenz s law will show that the induced current will be in the direction so that the induced B-field is in the upward direction. work out example 20.2
Application of Faraday s law Ground fault interrupters, used to protect against short circuits Pickups on electric guitars, converts the vibrations of the strings to an electrical signal.
Motional emf Earlier we changed the B-field with time. Now we keep the field constant. Look at the emf induced in a conductor moving through a magnetic field.
fig 20.12 Look at straight conductor of length (L) moving with constant velocity through a uniform B-field pointing into the page. In this example the velocity is normal to B-field. Force on the electrons (-) is F B = qvb downward Free electrons build up on the lower end, leaving a net positive charge on the upper end. This produces an E- field in the conductor. Electrons keep moving until the magnetic force is balanced out by the electric force qe. qe = qvb or E = vb
The potential difference across the conductor is given by V = E L so V = EL = BLv The upper end of the conductor will be at a higher potential than the lower end.
Now the conductor is part of a closed loop. See pictures on page 670. Conducting bar of length L slides along two fixed parallel conducting rails. Let the stationary part of the loop have a resistance R. A uniform and constant B-field is perpendicular to the plane of the loop. As the bar is pulled to the right, a magnetic force acts on the free charges in the bar. Since the bar is part of a closed loop, an induced current circulates.
The change in B and the induced current are produced from the change in area of the loop as the bar moves. If bar moved a distance x in time t, the flux changes as B. B =B A = BL x ( A = L x) using Faraday s Law with 1 loop (N = 1) and ignoring the direction for now B t BL x t BLv
We call this motional emf since it is produced from the motion of a conductor through a magnetic field. if we want to find the induced current, we use the resistance of the circuit R. V = IR I = V/R I R BLv R gives the magnitude of the induces current. Use Lenz s Law and right hand rule to get direction.
Lenz s Law revisited See pictures in book to practice getting the direction of the induced current. page 674 Look to see what is the direction of the change of flux. The induced current will produce flux in the opposite direction of the change in flux. http://micro.magnet.fsu.edu/electromag/java/le nzlaw/
Generators generators and motors operate on the principle of electromagnetic induction generator converts mechanical energy to electrical energy motor converts electrical energy to mechanical
Electrical generator Consists of a wire loop(s) rotated in a magnetic field. In a hydroelectric plant, falling water turns the blades of a turbine to rotate the loop. As the loop rotates, the magnetic flux through the loop changes with time. Changing flux induces an emf and a current. = NBA sin t the varies with time N = number of turns A = area or coil B = field strength = angular speed
Maximum emf from a generator occurs when t = = 90 0 or 270 0 = NBA This occurs when the plane of the loop is parallel to the magnetic field. f where f is the frequency of rotation In U.S. and Canada f = 60 Hz.
Motor is a generator acting in reverse. Current is supplied to a loop by some outside source. The magnetic torque on a carrying current loop, forces the loop to rotate. Remember from Ch. 19 that the torque on a current loop is: = B I A sin
Self Inductance Consider a circuit (current loop) with a switch, a resistor, and a source of emf (battery). When the switch is closed, the current does not change from 0 to its maximum value of I = /R immediately. Faraday s Law prevent this. As the current increases from zero, a magnetic field is produced and the flux through the loop increases. The increasing flux induces an emf that opposes the change in magnetic flux.
The net voltage across the resistor is the emf from the battery minus the induced emf. fig 20.24 The opposing emf results in a gradual increase in current. When the switch is closed, the current does not immediately drop to zero either. This effect is called self-inductance also see fig 20.25
Self Inductance Faraday s Law = -N B/ t The change in flux comes from the change in the current. = - L I/ t L = the inductance of the device If the current is increasing, the induced emf is negative.
Inductance Inductance has units of henry (H) 1 H = 1 V s/a These units can be found from the previous eqn. By setting equal the equations: = -N B/ t and = - L I/ t We can find: L N I B N I B
Inductance of a solenoid Al volume V V n Al n A l nl L nl N 2 0 2 0 2 0 ) ( l A N I AI l N I N L AI l N BA I l N ni B B B 2 0 2 0 0 0 0 ) (
example 20.7 Calculate inductance of solenoid of 300 turns and length 25 cm and area 4x10-4 m 2. L = 0N 2 A/length = (4 10 7 Tm / A) 300 2 (4x10 0.25m 4 m 2 ) L = 0.181 mh Calculate self induced emf in the solenoid if the current decreases at rate of 50 A/s. = -L I/ t = -0.181mH (-50 A/s) =9.05 mv (remember: H = V s/a)
RL circuits Series circuit with a resistor and an inductor. Resistor is used to limit the current flowing through the circuit. Resistance has units of ohms ( ). Inductor is a closely wrapped coil of many turns. Has an inductance. When current is turned on, it raises gradually to the maximum value of I = V/R V is the voltage provided by the battery. Save this for next segment of class. Will be on third exam.
Energy stored in a magnetic field Inductors store energy in a magnetic field. This is a potential energy. PE L = ½ L I 2 (L for inductance I for current) Note that this is similar to the expression for the energy stored in a capacitor (PE C = ½ CV 2 )
example 20.10 part b. 12 V battery is hooked up to a 25 resistor and a 5 H inductor. Find the max current: I max = V/R = (12V)/(25 ) = 0.48 A Find energy stored in the inductor when the current is maxed: energy = ½ (5 H)(0.48A) 2 = 0.576 J Worry about part c for later.